We present a polynomial-time algorithm for online differentially private synthetic data generation. For a data stream within the hypercube $[0,1]^d$ and an infinite time horizon, we develop an online algorithm that generates a differentially private synthetic dataset at each time $t$. This algorithm achieves a near-optimal accuracy bound of $O(t^{-1/d}\log(t))$ for $d\geq 2$ and $O(t^{-1}\log^{4.5}(t))$ for $d=1$ in the 1-Wasserstein distance. This result generalizes the previous work on the continual release model for counting queries to include Lipschitz queries. Compared to the offline case, where the entire dataset is available at once, our approach requires only an extra polylog factor in the accuracy bound.

Much of the research in differential privacy has focused on offline applications with the assumption that all data is available at once. When these algorithms are applied in practice to streams where data is collected over time, this either violates the privacy guarantees or results in poor utility. We derive an algorithm for differentially private synthetic streaming data generation, especially curated towards spatial datasets. Furthermore, we provide a general framework for online selective counting among a collection of queries which forms a basis for many tasks such as query answering and synthetic data generation. The utility of our algorithm is verified on both real-world and simulated datasets.

Differentially private synthetic data provide a powerful mechanism to enable data analysis while protecting sensitive information about individuals. However, when the data lie in a high-dimensional space, the accuracy of the synthetic data suffers from the curse of dimensionality. In this paper, we propose a differentially private algorithm to generate low-dimensional synthetic data efficiently from a high-dimensional dataset with a utility guarantee with respect to the Wasserstein distance. A key step of our algorithm is a private principal component analysis (PCA) procedure with a near-optimal accuracy bound that circumvents the curse of dimensionality. Different from the standard perturbation analysis using the Davis-Kahan theorem, our analysis of private PCA works without assuming the spectral gap for the sample covariance matrix.

High-dimensional multimodal data arises in many scientific fields. The integration of multimodal data becomes challenging when there is no known correspondence between the samples and the features of different datasets. To tackle this challenge, we introduce AVIDA, a framework for simultaneously performing data alignment and dimension reduction. In the numerical experiments, Gromov-Wasserstein optimal transport and t-distributed stochastic neighbor embedding are used as the alignment and dimension reduction modules respectively. We show that AVIDA correctly aligns high-dimensional datasets without common features with four synthesized datasets and two real multimodal single-cell datasets. Compared to several existing methods, we demonstrate that AVIDA better preserves structures of individual datasets, especially distinct local structures in the joint low-dimensional visualization, while achieving comparable alignment performance. Such a property is important in multimodal single-cell data analysis as some biological processes are uniquely captured by one of the datasets. In general applications, other methods can be used for the alignment and dimension reduction modules.

Motivated by biological considerations, we study sparse neural maps from an input layer to a target layer with sparse activity, and specifically the problem of storing $K$ input-target associations $(x,y)$, or memories, when the target vectors $y$ are sparse. We mathematically prove that $K$ undergoes a phase transition and that in general, and somewhat paradoxically, sparsity in the target layers increases the storage capacity of the map. The target vectors can be chosen arbitrarily, including in random fashion, and the memories can be both encoded and decoded by networks trained using local learning rules, including the simple Hebb rule. These results are robust under a variety of statistical assumptions on the data. The proofs rely on elegant properties of random polytopes and sub-gaussian random vector variables. Open problems and connections to capacity theories and polynomial threshold maps are discussed.

In recent literature, a general two step procedure has been formulated for solving the problem of phase retrieval. First, a spectral technique is used to obtain a constant-error initial estimate, following which, the estimate is refined to arbitrary precision by first-order optimization of a non-convex loss function. Numerical experiments, however, seem to suggest that simply running the iterative schemes from a random initialization may also lead to convergence, albeit at the cost of slightly higher sample complexity. In this paper, we prove that, in fact, constant step size online stochastic gradient descent (SGD) converges from arbitrary initializations for the non-smooth, non-convex amplitude squared loss objective. In this setting, online SGD is also equivalent to the randomized Kaczmarz algorithm from numerical analysis. Our analysis can easily be generalized to other single index models. It also makes use of new ideas from stochastic process theory, including the notion of a summary state space, which we believe will be of use for the broader field of non-convex optimization.

A long standing open problem in the theory of neural networks is the development of quantitative methods to estimate and compare the capabilities of different architectures. Here we define the capacity of an architecture by the binary logarithm of the number of functions it can compute, as the synaptic weights are varied. The capacity is an upper bound on the number of bits that can be "communicated" from the training data to the architecture over the learning channel. We study the capacity of layered, fully-connected, architectures of linear threshold neurons with $L$ layers of size $n_1,n_2, \ldots, n_L$ and show that in essence the capacity is given by a cubic polynomial in the layer sizes: $C(n_1,\ldots, n_L)=\sum_{k=1}^{L-1} \min(n_1,\ldots,n_k)n_kn_{k+1}$. In proving the main result, we also develop new techniques (multiplexing, enrichment, and stacking) as well as new bounds on the capacity of finite sets. We use the main result to identify architectures with maximal or minimal capacity under a number of natural constraints. This leads to the notion of structural regularization for deep architectures. While in general, everything else being equal, shallow networks compute more functions than deep networks, the functions computed by deep networks are more regular and "interesting".

We define the capacity of a learning machine to be the logarithm of the number (or volume) of the functions it can implement. We review known results, and derive new results, estimating the capacity of several neuronal models: linear and polynomial threshold gates, linear and polynomial threshold gates with constrained weights (binary weights, positive weights), and ReLU neurons. We also derive capacity estimates and bounds for fully recurrent networks and layered feedforward networks.

We define the capacity of a learning machine to be the logarithm of the number (or volume) of the functions it can implement. We review known results, and derive new results, estimating the capacity of several neuronal models: linear and polynomial threshold gates, linear and polynomial threshold gates with constrained weights (binary weights, positive weights), and ReLU neurons. We also derive capacity estimates and bounds for fully recurrent networks and layered feedforward networks.

The problem of Non-Gaussian Component Analysis (NGCA) is about finding a maximal low-dimensional subspace $E$ in $\mathbb{R}^n$ so that data points projected onto $E$ follow a non-gaussian distribution. Although this is an appropriate model for some real world data analysis problems, there has been little progress on this problem over the last decade. In this paper, we attempt to address this state of affairs in two ways. First, we give a new characterization of standard gaussian distributions in high-dimensions, which lead to effective tests for non-gaussianness. Second, we propose a simple algorithm, \emph{Reweighted PCA}, as a method for solving the NGCA problem. We prove that for a general unknown non-gaussian distribution, this algorithm recovers at least one direction in $E$, with sample and time complexity depending polynomially on the dimension of the ambient space. We conjecture that the algorithm actually recovers the entire $E$.